I really could not care less what Islamic doctrine says — flannel jesus

And vice versa. It doesn't care either what you think about it. Hence, the relationship is reciprocally perfectly sound.

Yah. I think that comes under the religious, rather the rational heading. — Vera Mont

I use rationality merely as a tool. I actually only use it when it suits me and I certainly do not identify with it. There may possibly be rational reasons to have hope and to keep faith in the future in spite of all adversity, in spite of all tribulations, but why use a hammer when the better tool is actually a screwdriver?

There are no rational reasons why life itself exists. So, why would there be a rational reason for insisting on staying alive and surviving in spite of all the hardships? When the going gets tough, the most straightforward solution is to seek out a crash course in spirituality and then use it to overcome your difficulties in life.

I personally do not believe that rationality can stimulate your survival instinct. In my opinion, it is simply of no use in that context.

Inescapable suffering that makes any joy in life impossible seems like a valid reason to me. — flannel jesus

According to Islamic doctrine, no suffering is inescapable. There is always hope.

Quran 4:30 And kill not yourselves. Surely, Allah is Merciful to you.

Patience is a virtue. Allah promises great rewards for those who bear hardship with patience.

If you have strong capacity to believe, then you can make use of such capacity to faith to believe that things will eventually get better and to sit out the temporary misery. That is the power of religious autosuggestion.

Not I, but Langendoen and Postal. If you wish you can take up the argument, I'm not wed to it, I'll not defend it here. I've only cited it to show that the case is not so closed as might be supposed from the Yanofsky piece. — Banno

Langendoen and Postal argue in "The vastness of natural languages", 1984, that natural-language sentences can be infinitely long.

https://aclanthology.org/J89-1006.pdf

This book is an extended argument in support of the theses that natural languages are transfinitely unbounded collections, that sentences are not limited in length (number of words) by any cardinal number, finite or transfinite, and that no constructive grammar can be an adequate grammar for any natural language.

https://fa.ewi.tudelft.nl/~hart/37/publications/the_papers/on_vastness.pdf

However, as I mentioned before, the authors do not so much argue for “not assuming a size law” but for “assuming the negation of a size law”. For example, the rules (if any) of English do not stipulate a maximum finite length of sentences; one can easily break such a stipulation by prefixing a maximum length sentence with “I know that”. The rules of English also do not explicitly state that sentences should be finite; one can add “All English sentence should be finite in length” to the rules or not. The authors argue, quite vociferously at times, against adding that condition mostly on the grounds that it is not a purely linguistic one. However, and this is where I disagree, they then conclude that, somehow, necessarily there should be sentences of infinite length.

Yanofsky, on the other hand, assumes that language sentences, especially predicate formulas that describe natural-number subset properties in ZFC, are necessarily finite.

Even though infinitary logics allows for infinitely long predicate formulas, they cannot be represented in language but only by their parse trees:

https://en.wikipedia.org/wiki/First-order_logic

Infinitary logic allows infinitely long sentences. For example, one may allow a conjunction or disjunction of infinitely many formulas, or quantification over infinitely many variables. Infinitely long sentences arise in areas of mathematics including topology and model theory.

Infinitary logic generalizes first-order logic to allow formulas of infinite length. The most common way in which formulas can become infinite is through infinite conjunctions and disjunctions. However, it is also possible to admit generalized signatures in which function and relation symbols are allowed to have infinite arities, or in which quantifiers can bind infinitely many variables. Because an infinite formula cannot be represented by a finite string, it is necessary to choose some other representation of formulas; the usual representation in this context is a tree. Thus, formulas are, essentially, identified with their parse trees, rather than with the strings being parsed.

Hence, the size of logic statements represented by language alone cannot be infinite. Therefore, the language of ZFC is still countably infinite.

Overcoming this constraint would require the use of meta-programs instead of predicates as set membership functions that have infinite while loops -- beyond primitive recursive arithmetic (PRA). These programs can then generate infinitely long predicates in the language of ZFC to describe Yanofsky's subsets. The use of such predicate-generating meta-programs instead of predicates as set membership functions is not supported in the language of ZFC.

Furthermore, this would still not help, because there are only countably infinite programs. There would still not be enough programs to describe all the uncountably infinite subsets of the natural numbers.

But the point is that "...the collection of all properties that can be expressed or described by language is only countably infinite because there is only a countably infinite collection of expressions" appears misguided, and at the least needs a better argument.

Your posts sometimes take maths just a little further than it can defensibly go. — Banno

The lemma that the number of possible expressions in language is countably finite is actually a core argument in Yanofsky's paper:

http://www.sci.brooklyn.cuny.edu/~noson/True%20but%20Unprovable.pdf

Another important uncountably infinite set is the collection of subsets of the natural numbers. The collection of all such subsets is uncountably infinite. Now that we have these different notions of infinity in our toolbox, let us apply them to our concept of true but unprovable statements. All language is countably infinite. The set of statements in basic arithmetic, the subset of true statements, and the subset of provable statements are all countably infinite.

This brings to light an amazing limitation of the power of language. The collection of all subsets of natural numbers is uncountably infinite while the set of expressions describing subsets of natural numbers is countably infinite. This means that the vast, vast majority of subsets of natural numbers cannot be expressed by language. The above examples of subsets of natural numbers are expressed by language, but they are part of the few rather than the many. The majority of the subsets are inexpressible. They defy language.

There is simple proof for the lemma that language is countably finite. Yanofsky's paper does not mention it but the proof is trivial:

https://math.stackexchange.com/questions/1206460/proving-that-the-set-of-all-english-words-is-countble

This is the question : Prove that the set of all the words in the English language is countable (the set's cardinality is אo) A word is defined as a finite sequence of letters in the English language.

Answer 1: There are 26 letters in the English language. Consider each letter as one of the digits on base 27. This mapping yields that the cardinality of your set is ≤|N|, hence this set is countable.

Answer 2: The set Sn of the English words with length n is finite (this is almost obvious). So it's also countable. Why is it finite? The set An of all sequences with length n made up of latin characters is finite as it contains 26n elements. Only some of these sequences are meaningful/actual English words. So Sn⊂An. So Sn is also finite. The set T for which you have to prove that it is countable is: T=S1∪S2∪S3∪... Now you have this theorem: "A countable union of countable sets is also countable". Applying it you get that T is also countable. Thus your statement has been proved.

But the whole point Wittgenstein's argument on the autonomy of mathematics systems is that a mathematical proposition is internally tied to its proof/proof system — Richard B

Wittgenstein's view is not compatible with model theory of which the core understanding is that the provability of a arithmetical proposition is tied to one system (PA) while its truth is tied to another system (ZFC).

https://en.wikipedia.org/wiki/Model_theory

In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the statements of the theory hold).

The model-theoretical approach is necessary in order to discover that there exists arithmetical truth in ZFC that is not tied to a proof in PA. Wittgenstein subscribes to a syntactic notion of truth. Model theory subscribes to the semantic nature of truth.

In fact, Wittgenstein does not properly distinguish between provability and truth.

If dealing with autonomous calculi then no matter how similar the rules of the two systems might be, as long as they differ - as long as we are dealing with distinct mathematical systems - It make no sense to speak of the same proposition occurring in each. The most that can be concluded is that parallel propositions occur in the two systems which can easily be mapped onto each other. — Richard B

Wittgenstein simply rejects the essence of model theory, because mapping propositions in PA to propositions in ZFC is exactly what model theory does. It is exactly about "the same proposition occurring in each".

Hence Godel was barred by virtue of the logical grammar of mathematical proposition from claiming that he had constructed identical versions of the same mathematical proposition in two different systems. — Richard B

In their seminal paper "On interpretations of arithmetic and set theory" of 2006, Richard Kaye and Tin Lok Wong formally proved that PA and ZF-inf are indeed bi-interpretable and therefore that the mapping in model theory -- in and of itself -- does not entail any of the risks that Wittgenstein incorrectly believes to exist.

Bi-interpretability:

- Every logic sentence in PA can be mapped to an equivalent logic sentence in ZF-inf. (Von Neumann)
- Every logic sentence in ZF-inf can be mapped to an equivalent logic sentence in PA. (Ackermann)

https://web.mat.bham.ac.uk/R.W.Kaye/publ/papers/finitesettheory/finitesettheory.pdf

The work described in this article starts with a piece of mathematical ‘folklore’ that is
‘well known’ but for which we know no satisfactory reference.

Folklore Result. The first-order theories Peano arithmetic and ZF set theory with the
axiom of infinity negated are equivalent, in the sense that each is interpretable in the
other and the interpretations are inverse to each other.

Perhaps the first and most obvious conclusion is that statements concerning the equiv-
alence of ‘Peano Arithmetic’ and ‘ZF with the axiom of infinity negated’ require some
care to formulate and prove. It is certainly true that PA and ‘ZF with the axiom of infin-
ity negated’ are equiconsistent for just about any sensible axiomatisation of the latter,
in the sense that interpretations exist in both directions.6 Probably this is the ‘folklore
result’ that most people remember. But for the finer result with interpretations inverse
to each other, careful axiomatisation of the set theory is required. A category theoretic
framework for interpretations is useful to direct attention to these refinements.

It is true that until 2006, model theory had always assumed the bi-interpretability of PA and ZF-inf without formal proof or similar investigation. Gödel never formally proved this mapping either. It was rather being considered self-evident. It would actually have been valid criticism to point out that Gödel assumed bi-interpretability without proof. For Wittgenstein to outright reject bi-interpretability, however, was clearly one bridge too far.

Why have you got it in for the math professors? — Tarskian

I don't. I also have no problems with bank staff. They are just pawns. They are just trying to make a living.

But why hate on the math professors? — fishfry

I don't hate on individual math professors. They are just pawns in the game.

One (or rather two) of the things I don't like, is the combo of academic credentialism combined with the student debt scam. Like all usury, it is a tool to enslave people. The banks conjure fiat money out of thin air and them want it back along with interest from teenagers who were lied to and most of whom will never have the ability to pay back. The ruling mafia even guarantees to the bankstering mafia that they will pay in lieu of the student, if he ultimately doesn't. First of all, though, they will exhaust all options afforded by the use of violent threats of lawfare.

These are all mathematical truths, but they're not very interesting mathematical truths. — fishfry

Agreed. Unpredictable truths in the physical universe are usually not particularly interesting either. The difference is that we can see them or at least observe them. That is why we know that the physical universe is mostly unpredictable. Our own eyes tell us. In order to "see" a mathematical truths, however, we need some written predicate. Otherwise, such truth is invisible to us.

If we completely ignore the unpredictable truths in the physical universe, it also gives us the impression of being beautifully and even majestically orderly. In that perception of the physical universe, there is no chaos. In that case, the physical universe also looks like a beautiful sculpture.

leaving only the beautiful sculpture that is modern mathematics — fishfry

Yes, and that is absolutely not the problem.

The problem is that people such as David Hilbert are convinced that the beautiful sculpture is all there is. Hilbert insisted on the idea that his colleagues had to work overtime in order to give him proof of his false belief:

https://en.wikipedia.org/wiki/Hilbert%27s_program

Statement of Hilbert's program

The main goal of Hilbert's program was to provide secure foundations for all mathematics.

Completeness: a proof that all true mathematical statements can be proved in the formalism.
Decidability: there should be an algorithm for deciding the truth or falsity of any mathematical statement.
...
Kurt Gödel showed that most of the goals of Hilbert's program were impossible to achieve.

The vast majority of people still see mathematical truth like Hilbert did. They still see mathematical truth as a predictable and harmonious orchestra of violins.

Out of the uncountably infinite and random universe of mathematical truth

Yes, you seem to know it perfectly fine. Most people, however, don't know it, simply because they don't want to know it.

They believe that one day we will discover the fundamental knowledge to see the entire physical universe also as a beautiful sculpture. We have already discovered the fundamental knowledge of arithmetic. Its axioms are known already and arithmetical truth is absolutely not a beautiful sculpture. Instead, it is uncountably infinite and random.

Right, and despite their work being concluded for quite some time now, people several times smarter than both you and I combined still hold math and science as tools of precision and meaningful discovery. — Philosophim

If you feel threatened by its chaotic nature, it means that it disturbs your ideological beliefs. Someone who really uses them as tools of precision and meaningful discovery would never feel threatened by that.

I find this point more interesting. Why? — Philosophim

It is probably best to use an example from the Soviet Union but in fact modern western society does exactly the same:

https://www.marxists.org/subject/marxmyths/john-holloway/article.htm

In speaking of Marxism as ‘scientific’, Engels means that it is based on an understanding of social development that is just as exact as the scientific understanding of natural development. For Engels, the claim that Marxism is scientific is a claim that it has understood the laws of motion of society. This understanding is based on two key elements: ‘These two great discoveries, the materialistic conception of history and the revelation of the secret of capitalistic production through surplus-value, we owe to Marx. With these two discoveries Socialism becomes a science. The claim that Marxism is scientific is taken to mean that subjective struggle (the struggle of socialists today) finds support in the objective movement of history. The notion of Marxism as scientific socialism has two aspects. In Engels’ account there is a double objectivity. Marxism is objective, certain, ‘scientific’ knowledge of an objective, inevitable process. Marxism is understood as scientific in the sense that it has understood correctly the laws of motion of a historical process taking place independently of men’s will. All that is left for Marxists to do is to fill in the details, to apply the scientific understanding of history. The attraction of the conception of Marxism as a scientifically objective theory of revolution for those who were dedicating their lives to struggle against capitalism is obvious. At the same time, however, both aspects of the concept of scientific socialism (objective knowledge, objective process) pose enormous problems for the development of Marxism as a theory of struggle.

It is very convincing, because it sounds scientific, and because it insists that it is scientific, and especially because you will get burned at the Pfizer antivaxxer stake if you refuse to memorize this sacred fragment from the scripture of scientific truth for your scientific gender studies exam.

As you can see, everybody who craves credibility insists on sailing under the flag of scientism and redirect the worship and adulation of the masses for the omnipotent powers of science to themselves and their narrative.

Again, hyperbole. I can assure you if we were able to predict how everything in the universe worked — Philosophim

The complete and perfect theory of everything cannot do that. It won't be able to predict everything. It would improve our ability to predict the physical universe from 0.1% to 0.3% of the true facts. So, it will possibly triple the predictive power of physics but not more than that.

We already have the theory of everything for the natural numbers, which is PA. It does not help us to predict the vast majority of mathematical truths. Most of the truth about the natural numbers is still unpredictable.

No, nothing you have discovered here has shaken the foundations of math or science. — Philosophim

I did not discover anything. Gödel certainly did. Chaitin also did. Yanofsky moderately did. I just mentioned their work.

What method did you use to find out that its true? — Philosophim

It is an opinion and not a theorem. There is nothing wrong with mathematics or with science. My problem is with positivism and scientism. I find these ideological beliefs to be very dangerous.

The case for a higher authority, an absolute authority, has to be argued philosophically. Not religiously, that is, not according anything so instantly assailable. — Constance

Well, Christianity is indeed collapsing. Ever more rapidly.

Christianity has indeed turned out to be assailable but certainly not easily or instantly. It took centuries until the French Revolution for its assailants to finally make a dent. The other religions are still doing fine. I think that it has become clear that it is not possible to dislodge them. It is not possible to convince a traditional Jew out of Judaism or a traditional Muslim out of Islam.

We just don't have time to figure out alternative solutions to religion. If you don't have something handy that works right now, and that already has a track history of success, then you are going to be too late to still make a difference. Life moves on. Life is also short. I cannot wait for a solution to fall out of the skies. In fact, it has already fallen out of the sky. So, why not just use it?

It sounds as though you yourself hold some rather specific and rigid beliefs that likewise are not entirely objective in their genesis. — Pantagruel

Well, yeah, I rigidly believe that we should not give powers to people that only Allah should have, and if Allah does even not exist, then so much the better.

Just like I wouldn't grab a wrench if I were studying the atomic level of the universe, one shouldn't use certain language and terms when dealing with the foundations of knowledge and mathematics. — Philosophim

The true nature of the universe of mathematical facts makes lots of people uncomfortable.

Imagine that we had a copy of the theory of everything?

It would allow us to mathematically prove things about the physical universe. It would be the best possible knowledge that we could have about the physical universe. We would finally have found the holy grail of science.

What would the impact be?

Well, instead of being able to predict just 0.1% of the facts in the physical universe, this would improve to something like 0.3%; and not much more.

Scientism is widespread as an ideology in the modern world. Any true understanding of the nature of mathematical truth deals a devastating blow to people who subscribe to it. This is exactly why I like this subject so much.

The hyperbole just isn't true. — Philosophim

That is wishful thinking.

You may not want it to be true, but it is.

In 1931, Gödel's incompleteness theorems dealt a major blow to positivism and scientism, but it was just the beginning. It is only going to keep getting worse. As Yanovsky writes in his paper:

Gödel’s famous incompleteness theorem showed us that there is a statement in basic arithmetic that is true but can never be proven with basic arithmetic. That is just the beginning of the story.

In my opinion, scientism needs to get attacked and destroyed because its narrative is not just arrogant but fundamentally evil. It is a dangerously false pagan belief that misleads its followers into accepting untested experimental vaccine shots from the lying and scamming representatives of the pharmaceutical mafia; and that is just one of the many examples of why all of this is not hyperbole.

it took other people to fix the inconsistency in his proof just to then generate further issues in these updated proofs. — Lionino

Modal collapse is not an inconsistency. Who told you that?

It just means that the proof reverts to standard non-modal logic.

Since non-modal logic is the default logic anyway, does that mean that pretty much all proofs in mathematics are inconsistent?

In modal logic, modal collapse is the condition in which every true statement is necessarily true, and vice versa; that is to say, there are no contingent truths, or to put it another way, that "everything exists necessarily".

Since standard logic does not even distinguish between necessary and contingent truth, what is supposedly the big problem?

Furthermore, Anderson has fixed the issue and removed the modal collapse. This is not essential at all. It is just nice to have and not more than that.

In fact, it may even be a good thing. It means that the proof works, even without using modal modifiers. So, the proof would be valid, even in plain, standard logic.

And everytime when someone makes an universal statement that ought to apply to everything, watch out! — ssu

That may very well be in violation of Carnap's diagonal lemma:

"For each property of logic sentences, there exists a true sentence that does not have it, or a false sentence that does."

But then again, it still needs to be a property of logic sentences. For example, a property of natural numbers can apply to all natural numbers.

I think you’re mis-using the word there. If everything were chaotic, nothing would exist, and if everything were perfectly ordered, nothing would change. Existence requires both. Beyond that, I can’t see the point, if there is one. — Wayfarer

People seem to understand this about the truth in the physical universe. They tend to reject this about the truth in arithmetic. I wanted to point out that the situation is the same.

Hippasus, who found irrational numbers was ostracized and when he drowned at sea, it was the "punishment of the Gods". — ssu

Yes, these people want a kind of certainty that simply does not exist ...

The existential claim carries the onus probandi — above

Gödel did exactly that. He provided a mathematically unobjectionable proof. Of course, math never does more than advertised. The witness for the existential theorem has successfully been supplied. Next.

Out of interest, what type of believer are you? Muslim or Christian, or something less specific? — Tom Storm

Originally born a Catholic. In the meanwhile, I came to the conclusions that Christians no longer intend to use the rules in the scripture as a benchmark to assess societal sanity. So, my sympathies are definitely much more Muslim nowadays. So, the problem is not necessarily Christianity but the lack of enthusiasm of the Christians. But then again, they completely mishandled the reformation too. The following was clearly not the solution either:

Charles V's "Edict of Blood" of 1550 in the Burgundian Netherlands

No one shall print, write, copy, keep, conceal, sell, buy or give in churches, streets, or other places, any book or writing made by Martin Luther, John Oecolampadius, Huldrych Zwingli, Martin Bucer, John Calvin, or other heretics reprobated by the Holy Church.
...
That such perturbators of the general quiet are to be executed, to wit: the men with the sword and the women to be buried alive, if they do not persist in their errors; if they do persist in them, then they are to be executed with fire; all their property in both cases being confiscated to the crown.

This approach failed in the Burgundian Netherlands but it actually succeeded in France. After successfully eradicating the reformation in France, the Catholic Church probably thought that they were good to go, only to later on end up with the French revolutionaries who did not even try to reform the religion but got rid of it altogether. Forcing everybody to join your club is clearly not a good idea.

Perhaps some can see this as chaotic, but math itself is quite logical and hence quite orderly. Unprovability or uncomputability doesn't mean chaotic. Math is orderly, we just have limitations on what to compute or prove. — ssu

I started using unpredictability as somewhat a synonym for unprovability because of how Stephen Hawking put it:

https://www.hawking.org.uk/in-words/lectures/godel-and-the-end-of-physics

What is the relation between Godel’s theorem and whether we can formulate the theory of the universe in terms of a finite number of principles? One connection is obvious. According to the positivist philosophy of science, a physical theory is a mathematical model. So if there are mathematical results that can not be proved, there are physical problems that can not be predicted.

So, we are sitting on a system that is largely unpredictable because most of its truths are unprovable. A system that is largely unpredictable is deemed chaotic:

Chaos theory is an interdisciplinary area of scientific study and branch of mathematics. It focuses on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions. These were once thought to have completely random states of disorder and irregularities.[1] Chaos theory states that within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnection, constant feedback loops, repetition, self-similarity, fractals and self-organization.[2]

This can happen even though these systems are deterministic, meaning that their future behavior follows a unique evolution[8] and is fully determined by their initial conditions, with no random elements involved.[9] In other words, the deterministic nature of these systems does not make them predictable.[10][11] This behavior is known as deterministic chaos, or simply chaos.

The only minor difference between the universe of arithmetical truth and a chaotic system is that there are no "initial conditions" that we can change in order to produce a completely different version of arithmetical truth.

Even if it obviously c is a natural number and has a precise point on the number line, not some range, we cannot prove c exactly. — ssu

Imagine that we somehow have the information that c=17. Without additional information, it is not possible to prove it. In that sense, c is true but not provable. It could even be impossible to prove. In the standard model of arithmetic, i.e. in the natural numbers, we can somehow see that c=17 but in various nonstandard models, we can see that c is not 17. In those circumstances, proof is not even possible. Only when c=17 in all models of arithmetic, a proof is possible.

The problem rises because we just assume that everything in math has to be provable. — ssu

Yes, David Hilbert even wanted proof for that. In his view, every true statement must have a proof:

https://en.wikipedia.org/wiki/Hilbert%27s_program

Statement of Hilbert's program

- A formulation of all mathematics; in other words all mathematical statements should be written in a precise formal language, and manipulated according to well defined rules.

- Completeness: a proof that all true mathematical statements can be proved in the formalism.
- Consistency: a proof that no contradiction can be obtained in the formalism of mathematics. This consistency proof should preferably use only "finitistic" reasoning about finite mathematical objects.
- Conservation: a proof that any result about "real objects" obtained using reasoning about "ideal objects" (such as uncountable sets) can be proved without using ideal objects.
- Decidability: there should be an algorithm for deciding the truth or falsity of any mathematical statement.

Hilbert believed it so strongly that he insisted that all his colleagues should work on proving the above. A lot of people still believe it. You can give them proof that it is absolutely impossible, but they simply don't care about that. They will just keep going as if nothing happened. You can't wake a person who is pretending to be asleep.

The existential claim carries the onus probandi (generally, existential claims are verifiable and not falsifiable, universal claims are falsifiable and not verifiable), it's not for someone else to disprove. — jorndoe

Since we are talking about proof, it is the mathematical view on the subject that matters. Everybody else should avoid using the term ¨proof¨. What they produce as justification, is at best "evidence". It is never proof.

Existential proofs are much easier to produce than impossibility proofs. Gödel successfully produced one. It does require higher-order modal logic, but that is still trivially simple compared to what impossibility proofs typically rest on.

If you want to prove an impossibility, you need to painstakingly discover and make use of a structural constraint that will successfully reject every possible witness. In absence of such structural constraint, you would need omniscience.

There are impossibility proofs. For example, Abel-Ruffini theorem rests on the Galois correspondence as a structural constraint, while Fermat's last theorem rests on the modularity theorem. So, it is possible. There are impossibility proofs, but non-trivial ones typically took centuries to discover.

Therefore, you probably understand now that impossibility is not the default in mathematics. On the contrary, it is the result of centuries of hard work. Gödel successfully did his work and produced an existential proof. Where can we see the commendable mathematical work produced by an atheist in which he supports his impossibility claim?

By the way, atheists really need to prove that they are not making use of omniscience for their impossibility claim that an omniscient entity does not exist. This burden is on them and not on us.

This is a far cry from the point that math can be difficult to put into words. The proof is in the very fact you're able to post online consistently for us to read your posts. That was all capable through math. — Philosophim

All of this is the result of using just one direction ("soundness"):

If it is provable, then it is always true.

That is the only direction that we use in engineering. We never use the other direction:

If it is true, then it is pretty much never provable. It is a rare exception, if it is.

In math, we mostly don't even see these unpredictable truths. How would we? In the physical universe, we can definitely see the unpredictable chaos, but we mostly ignore it. Mathematical truth is as chaotic as the truth in the physical universe. In my opinion, there is not much difference. We typically just don't want to know about it.

Nevertheless, and to all practical purposes, mathematics enables a very wide range of successful predictions, doesn’t it? The mathematical physics underlying the technology on which this conversation is being conducted provides a high degree of prediction and control, doesn’t it? Otherwise, it wouldn’t work. — Wayfarer

There are two directions.

If it is provable, then it is always true. (aka, soundness theorem) In this direction, everything is very orderly. That is the only direction that we really use. That is why works so well.

If it is true, then it is almost surely not provable. In this direction, everything is very chaotic. We almost never use this direction. In fact, we cannot even see most of these random truths. So, why would we try to prove them?

It took Gödel all kinds of acrobatics in metamathematics to discover that these unpredictable truths even exist.

Before the publication of Gödel's paper in 1931, nobody even knew about these random truths. Most mathematicians were actually convinced that if it is true, then it is surely provable. Pretty much everybody on the planet was wrong about this before 1931. They were all deeply steeped in positivism. David Hilbert even asked for a formal proof of this glaring error. In fact, there are still a lot of people who believe this. Almost a century after its refutation, it is still a widespread misconception.

For the indifferent or one who finds the question incoherent it is not a matter of truth value, and that is the point. So, Joshs "none of the above": seems most apt. — Janus

That point of view is not a problem.

Only a 'yes' or 'no' answer constitutes a real commitment.

For 'yes' answer, you need to locate a constructive witness. This is possible. Gödel did exactly that. For 'no' answer, the default situation is that you generally need omniscience.

In fact, impossibility proofs do exist. They are not completely impossible. However, they typically require discovering a structural constraint that could never be satisfied by any possible witness.

A good example is the Abel-Ruffini theorem. There is no solution in radicals to general polynomial equations of degree five or higher. It took centuries to prove this because at first glance it requires omniscience. It required discovering the Galois correspondence as a structural constraint that any solution would violate. Fermat's last theorem is another good example. Without the modularity theorem, it would also require omniscience to prove this impossibility. It took over 350 years to pull off the proof.

Where is the structural constraint that makes a "no" answer to the "Does God exist?" question viable without requiring omniscience? Proving an impossibility is substantially harder than locating a suitable witness for a theorem. That is why a proof for atheism is several orders of magnitude more unlikely than a proof for religion.

I believe Chaitin made a similar point. He has a proof of Gödel's incompleteness theorems from algorithmic complexity theory. I believe he says that mathematical truth is essentially random. Things are true just because they are, not because of any deeper reason.

This sounds related to what you're saying. — fishfry

Yes, Yanofsky's paper also mentions Chaitin's work:

http://www.sci.brooklyn.cuny.edu/~noson/True%20but%20Unprovable.pdf

Gregory Chaitin described an innovative way of finding true but unprovable statements. He started by examining the complexity of the axioms of a logical system. He showed that there are certain statements that are much more complex than the axioms of the system. Such statements are true but cannot be proven by the axioms of the logical system. The following motto is sometimes used to explain this:

“A fifty-pound logical system cannot prove a seventy-five-pound theorem.”

In particular, basic arithmetic is a logical system that has a level of complexity and so there are certain types of statements that are true but too complex to be proven using basic arithmetic. The main point for our story is that within basic arithmetic we can always find more complicated statements of a certain type. Hence, there are infinitely many true but unprovable statements.

Cristian Calude extended Chaitin’s findings. He demonstrated that provable statements are actually very rare within the space of all true statements. In a sense, he showed that in the space of all true statements, every provable true statement is surrounded by many unprovable true statements.

This means that most (but not all) mathematical truth is essentially random.

Yanofsky's paper mentions an even larger class of random mathematical truth: unprovable because ineffable ("inexpressable"). There is no way to prove truths that cannot even be expressed in language. Because in that case, how are you going to express the proof? That class of random truths is even larger than Chaitin's random truths.

But then again, there exists a small class of true and provable statements.

In fact, nature of mathematical facts is quite similar to the nature of facts in the physical universe. Mostly random but with a relatively small class of facts that is still predictable. Unlike what most people believe, math is not more orderly than the physical universe itself.

In Godel, it appears consistency is assumed — tim wood

No, Gödel does not assume consistency. In Gödel's theorems, consistency is exactly the question. In mathematics we implicitly assume consistency. In metamathematics, we don't.

Why should we suppose that natural languages are only countably infinite? — Banno

You can enumerate every sentence in natural language in a list. Therefore, it maps one to one onto the natural numbers. Therefore, their set is countable.

https://en.wikipedia.org/wiki/Countable_set

In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers.[a] Equivalently, a set is countable if there exists an injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural number, or that the elements of the set can be counted one at a time, although the counting may never finish due to an infinite number of elements.

For natural language to be uncountable, you must find a sentence that cannot be added to the list. To that effect, you would need some kind of second-order diagonal argument.

Could you say a little more about what makes an unprovable mathematical proposition true? — Joshs

The fact that we can prove that it exists.

Let's start from Carnap's diagonal lemma. In the context of Peano arithmetic (PA), for each property φ(n) accepting one natural number n as input argument, there exists a true sentence S that does not have the property or a false sentence S that does have it:

PA ⊢ ∀ φ ∃ S ( S ⇔ ¬φ(ÍSÎ ) )

This is, in fact, the only hard part in Gödel's proof. The proof for the lemma is very short but it is widely considered to be incomprehensible:

https://proofwiki.org/wiki/Diagonal_Lemma

Say that Bew(ÍSÎ) is a property in PA that is true if it proves S and false when it doesn't. In that case, the lemma applies:

PA ⊢ ∀ φ ∃ S ( S ⇔ ¬Bew(ÍSÎ ) )

There exists a true sentence that is not provable or a false sentence that is provable. Hence, PA is incomplete or inconsistent. Let's denote this sentence as G:

PA ⊢ G ⇔ ¬Bew(ÍGÎ ) )

So, now we have a sentence that is (true or unprovable) or (false and provable). In fact, G is also a truly constructive witness for the theorem. But then again, we do not even need this particular sentence, because in the meanwhile, we also have Goodstein's theorem that is true but unprovable in PA:

https://en.wikipedia.org/wiki/Goodstein%27s_theorem

It is very hard to discover this kind of true but unprovable sentences. But then again, we also know that they massively outnumber the true and provable sentences. True but unprovable is the rule while true and provable is the exception. This is the paradoxical situation of the truth in PA. The truth in PA is highly chaotic but it is very hard for us to see that.

He attributes to Godel this idea:
:“'Basic arithmetic cannot prove a contradiction.' — tim wood

The paper actually says:

As a bonus, Gödel described another interesting statement in the language of basic arithmetic. He was able to formulate a statement in basic arithmetic that says:

“Basic arithmetic cannot prove a contradiction.”

So, it only insists that the sentence "PA cannot prove a contradiction" can be expressed in PA itself.

In the following paper, containing a version of the proof, the author expresses it by reifying the truth value for falsehood (⊥):

http://sammelpunkt.philo.at/id/eprint/2676/1/Bagaria.pdf

page 12:

Let CON(T) be the sentence ¬BewT(Í⊥Î). Thus, CON(T) says, via coding, that T is consistent.

In another paper, with another version of the proof, the author insist that it is enough to express the unprovability of any arbitrary falsehood. No need to reify truth values:

http://www.sfu.ca/~kabanets/308/lectures/lec11.pdf

We say that a proof system P is consistent if P does not prove both A and ¬A for some sentence A. That is, a consistent proof system cannot derive a contradiction A ∧ ¬A. In the case of a proof system P for arithmetic, we get that P is consistent iff P does not prove the sentence “1 = 2” (since 1 6 = 2 can be derived in P by the usual axioms (of Peano arithmetic) for the natural numbers).

ConsP : “the sentence “1 = 2” is not provable in P ”

So, in that case, let's consP be the sentence ¬BewP(Í1 = 2Î).

But then again, it is also perfectly possible to express the notion of consistency in full -- straight from its definition -- that PA does not prove both A and ¬A for all sentences A of PA:

PA ⊢ ∀ A ( ¬Bew(ÍA ∧ ¬AÎ ) )

In all cases, regardless of how you express consistency of PA, the proof for the second incompleteness theorem always proceeds by considering the first incompleteness theorem:

PA ⊢ ∃ A ( A ⇔ ¬Bew(ÍAÎ ) )

The above means: There exists a sentence A that is (true and not provable) or (false and provable).

Say that G is such sentence:

PA ⊢ G ⇔ ¬Bew(ÍGÎ )

If PA can prove its consistency, then it can obviously also prove that G is consistent:

PA ⊢ ¬Bew(ÍG ∧ ¬GÎ )

By using the Hilbert-Bernays rewrite rules -- with a few more steps -- we can then prove that this expression leads to the following contradiction about G:

PA ⊢ ¬(G ⇔ ¬Bew(ÍGÎ ))

There are many ways to formulate the consistency of PA, i.e. Cons(PA), but proving it will always lead to a contradiction. Therefore, Cons(PA) is unprovable. According to the first incompleteness theorem, the following sentence is true:

Cons(PA) ∨ Incompl(PA)

So, PA is inconsistent or incomplete. However, we do not know if Cons(PA) is true. We can only come to that conclusion by proving it, but how are we supposed to do that? So, I disagree with the author when he writes:

It turns out that this statement is also true but unprovable.

This statement is only unprovable.

Of course, we can use Gentzen's equiconsistency proof with PRA but that does not prove PA's consistency. It just proves that it is equiconsistent with PRA (primitive recursive arithmetic). Who says that PRA is consistent? We don't know that. Other authors sometimes write that we can prove PA's consistency from within ZFC. Fine, but who says that ZFC is consistent?

Hence, we can only assume PA's consistency. We cannot just state Cons(PA) to be true. This cannot be done.

So you would have 'don't care' mapped to unknown? — Tom Storm

Well, how many additional truth values do we need to invent before all our needs for additional truth values will have been completely satisfied?

Seriously, it is a slippery slope. We are going to end up with more truth values than genders!

perhaps due simply to a complete lack of interest — Janus

If someone is not interested in the issue, fine, but then his answer should still get mapped to the truth value unknown/maybe.

There is no need for an additional truth value to reflect this.

Again, the answer unknown/maybe is perfectly fine. Unlike the answer "no", it does not reflect a problem of omniscience.

Contrary to what the weekly sophist implies, choice of axioms is not arbitrary. — Lionino

That is clearly a straw man. You are attacking an argument that I did not make. You are using Don Quichotte tactics. Who exactly is the sophist here?

As previously stated, you have not read the article you yourself linked. Congrats. — Lionino

That is classical non sequitur. Again some word-salad nonsense.

Godel flawlessly proved the equiconsistency between his theorem and the axioms from which it follows. Godel's proof is therefore mathematically unobjectionable. Of course, Godel did not prove the axioms themselves. But then again, he is not even supposed to.

Your arguments amount to just a bit of black mouthing and shit talking. That says much more about you than about Godel's work.

But religion is certainly not about this. It is about ethics. What is ethics? — Constance

What are sound ethics? We take a snapshot of a sane society as well as an inventory of its rules. This is a suitable benchmark for ethical sanity. Benchmarking sane rules when it was still possible is what the scriptures implicitly do.

That is why the scriptures had to be transmitted as soon as possible, in order to front run the degeneracy that would inevitably follow later on. That is also why it is no longer possible to transmit new scriptures. It would simply describe the depravity of our contemporary society and not be suitable as a benchmark. Prophetic times are over.

Thanks to the scriptures we still know what we are supposed to be and how we are supposed to behave. It is a fantastic tool against the manipulative narrative of the ruling mafia. They handsomely benefit from growing depravity. We don't.

Well,

PA is a chaotic complex system without initial conditions. — Tarskian

looks a bit... overstated. — Banno

I have just found an interesting paper that elaborates on why the overwhelming majority of true statements in arithmetic are unprovable -- and therefore unpredictable. In fact, most truth in PA is simply ineffable.

http://www.sci.brooklyn.cuny.edu/~noson/True%20but%20Unprovable.pdf

We have come a long way since Gödel. A true but unprovable statement is not some strange, rare phenomenon. In fact, the opposite is correct. A fact that is true and provable is a rare phenomenon. The collection of mathematical facts is very large and what is expressible and true is a small part of it. Furthermore, what is provable is only a small part of those.

Most mathematical truth is unpredictably chaotic.

That the conclusions follow from the premises can be said about every fiction book — Lionino

Well no. You need to be quite sure that the book is about a sane society. You cannot just invent one. It needs to have historically existed.

If you had actually read the "article" you linked, you would know that Gödel's original axioms are inconsistent — Lionino

They are not inconsistent. There may be an issue of modal collapse but Curtis Anderson proposed a fix for that. It is not a major problem.

Aren't these the "initial conditions"...? These are the Peano axioms:
Zero is a natural number.
Every natural number has a successor in the natural numbers.
Zero is not the successor of any natural number.
If the successor of two natural numbers is the same, then the two original numbers are the same.
If a set contains zero and the successor of every number is in the set, then the set contains the natural numbers.
It's far from obvious what this has to do with chaotic systems.

I'm not following Tarskian's argument at all. — Banno

A chaotic system is one that follows a seemingly random path albeit deterministic. If you repeat the path with exactly the same initial conditions, it will follow exactly the same path.

Example:

Initial condition: "hello world"
sha256 hash: b94d27b99...
sha256 hash: 049da0526...
and so on (you keep feeding the output as new input)

If you change one letter to the initial seed, the path will change completely.

This is a chaotic complex system. Its facts look random. If you don't know the initial seed, it is for all intents and purposes effectively random.

Since most facts in arithmetic (PA), i.e. the arithmetical truth, are unprovable from the axioms, it has similar characteristics to the example system.

However, there is no initial seed in PA. The chaos in PA is caused by another phenomenon. Provable statements in PA are not merely true in the model/universe of the natural numbers. They are also true in an unlimited number of nonstandard models/universes of arithmetic. Most of its true facts are, however, not true in all its models/universes. That would be a precondition for their provability/predictability. That is why most facts in arithmetic are not predictable/provable from theory.

https://en.m.wikipedia.org/wiki/Non-standard_model_of_arithmetic

In mathematical logic, a non-standard model of arithmetic is a model of first-order Peano arithmetic that contains non-standard numbers. The term standard model of arithmetic refers to the standard natural numbers 0, 1, 2, …. The elements of any model of Peano arithmetic are linearly ordered and possess an initial segment isomorphic to the standard natural numbers. A non-standard model is one that has additional elements outside this initial segment. The construction of such models is due to Thoralf Skolem (1934).

This phenomenon explains why PA is incomplete (i.e. having unprovable truths) or inconsistent (i.e. provable falsehoods) or even possibly both.

Hence, the nature of the majority of facts in arithmetic is chaotic, i.e. unpredictable (unprovable).

might have been done by any number of fanatics (Castro, Hitler, Putin, whoever) — Tom Storm

Hitler tried and failed.

The effectiveness of math can be demonstrated through its consistency and predictability. — Tom Storm

Concerning the consistency of any theory such as PA (Peano arithmetic theory), it is merely an assumption. Gödel's second incompleteness theorem proves that if a mathematical system is capable of proving its consistency, it is necessarily inconsistent.

Therefore, the consistency of PA is based on faith alone.

Of course, we use PA to maintain consistency in downstream applications, and it works surprisingly well, but it is certainly not a provable property of PA.

Concerning the predictability of PA, whenever there exist true but unprovable theorems in a system, they massively outnumber the provable ones.

Hence, PA is mostly unpredictable.

According to Stephen Hawking, the unpredictability of the universe is tightly connected to the unpredictability of PA:

https://www.hawking.org.uk/in-words/lectures/godel-and-the-end-of-physics

What is the relation between Godel’s theorem and whether we can formulate the theory of the universe in terms of a finite number of principles? One connection is obvious. According to the positivist philosophy of science, a physical theory is a mathematical model. So if there are mathematical results that can not be proved, there are physical problems that can not be predicted.

PA is predictable in one direction, with provable implying true. However, when you look at the universe of true facts in PA, it is not predictable, because true rarely implies provable. PA is highly chaotic, albeit in a deterministic way.

https://en.wikipedia.org/wiki/Chaos_theory

Chaos theory is an interdisciplinary area of scientific study and branch of mathematics. It focuses on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions. These were once thought to have completely random states of disorder and irregularities.[1] Chaos theory states that within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnection, constant feedback loops, repetition, self-similarity, fractals and self-organization.[2]

PA is a chaotic complex system without initial conditions.

even within the single religion. It is unpredictable and inconsistent.

PA is also mostly unpredictable and its consistency is at best a statement of faith.

What do you mean by the term "existence"? — 180 Proof

https://en.wikipedia.org/wiki/Existential_quantification

Existential quantification

In predicate logic, an existential quantification is a type of quantifier, a logical constant which is interpreted as "there exists", "there is at least one", or "for some". It is usually denoted by the logical operator symbol ∃, which, when used together with a predicate variable, is called an existential quantifier ("∃x" or "∃(x)" or "(∃x)").

You are unhappy with students being taught the state of the art in their field. — fishfry

It is not the state of the art in their field.

The president of the United States draws a paycheck. — fishfry

Not because he wants to. There are so many people willing to pay a million dollars just for an 15-minute appointment with him, but he is not allowed to accept the money. He could easily put it up for auction. So, the paycheck is just part of a dog-and-pony show. Money does not matter to the people who print the money.

All the 18 year olds are apprenticed out to people who will pay them even though they're completely ignorant? You can't be serious. What are you talking about? — fishfry

That is how it used to work. They would become apprentices at the age of 14 and learn a job. This makes much more sense than keeping them in holding pens like cattle. There was no youth unemployment in past times.

So maybe you're against large organizations. — fishfry

Not necessarily.

I have worked as a contractor and done lots of consulting work at large organizations.

I would never have wanted to be staff, though. When we talk about "bottom line", the only one that mattered to me was my own "bottom line". I was not interested in selflessly "sacrifice" myself for someone else's bottom line. I cannot identify with the profit of the company. I can only identify with my own profit. I understand that C-level execs somewhat care, since they receive payments for when the stock goes up, but the other salaried office drones? Seriously, why would anybody else care?